Standard +0.8 This is a substantial multi-part Further Maths question requiring: (1) finding a normal vector via cross product to convert parametric to Cartesian form, (2) finding line of intersection of two planes by solving simultaneously, and (3) applying both distance and angle formulas with algebraic manipulation to find two unknowns. While each technique is standard for FM students, the combination of multiple steps, careful vector arithmetic, and solving the final system pushes this above average difficulty.
The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) + \mu ( - \mathbf { i } + \mathbf { k } )\). Obtain a cartesian equation of \(\Pi _ { 1 }\) in the form \(p x + q y + r z = d\).
The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) = 12\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
The line \(l\) passes through the point \(A\) with position vector \(a \mathbf { i } + ( 2 a + 1 ) \mathbf { j } - 3 \mathbf { k }\) and is parallel to \(3 c \mathbf { i } - 3 \mathbf { j } + c \mathbf { k }\), where \(a\) and \(c\) are positive constants. Given that the perpendicular distance from \(A\) to \(\Pi _ { 1 }\) is \(\frac { 15 } { \sqrt { } 6 }\) and that the acute angle between \(l\) and \(\Pi _ { 1 }\) is \(\sin ^ { - 1 } \left( \frac { 2 } { \sqrt { } 6 } \right)\), find the values of \(a\) and \(c\).
The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) + \mu ( - \mathbf { i } + \mathbf { k } )$. Obtain a cartesian equation of $\Pi _ { 1 }$ in the form $p x + q y + r z = d$.
The plane $\Pi _ { 2 }$ has equation $\mathbf { r } . ( \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) = 12$. Find a vector equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
The line $l$ passes through the point $A$ with position vector $a \mathbf { i } + ( 2 a + 1 ) \mathbf { j } - 3 \mathbf { k }$ and is parallel to $3 c \mathbf { i } - 3 \mathbf { j } + c \mathbf { k }$, where $a$ and $c$ are positive constants. Given that the perpendicular distance from $A$ to $\Pi _ { 1 }$ is $\frac { 15 } { \sqrt { } 6 }$ and that the acute angle between $l$ and $\Pi _ { 1 }$ is $\sin ^ { - 1 } \left( \frac { 2 } { \sqrt { } 6 } \right)$, find the values of $a$ and $c$.
\hfill \mbox{\textit{CAIE FP1 2010 Q12 OR}}